direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C5×D27, C27⋊C10, C135⋊2C2, C45.2S3, C15.2D9, C9.(C5×S3), C3.(C5×D9), SmallGroup(270,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C27 — C5×D27 |
Generators and relations for C5×D27
G = < a,b,c | a5=b27=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C5×D27
►On 135 points
Generators in S135
(1 129 85 75 28)(2 130 86 76 29)(3 131 87 77 30)(4 132 88 78 31)(5 133 89 79 32)(6 134 90 80 33)(7 135 91 81 34)(8 109 92 55 35)(9 110 93 56 36)(10 111 94 57 37)(11 112 95 58 38)(12 113 96 59 39)(13 114 97 60 40)(14 115 98 61 41)(15 116 99 62 42)(16 117 100 63 43)(17 118 101 64 44)(18 119 102 65 45)(19 120 103 66 46)(20 121 104 67 47)(21 122 105 68 48)(22 123 106 69 49)(23 124 107 70 50)(24 125 108 71 51)(25 126 82 72 52)(26 127 83 73 53)(27 128 84 74 54)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27)(28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54)(55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81)(82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108)(109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135)
(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)(13 16)(14 15)(29 54)(30 53)(31 52)(32 51)(33 50)(34 49)(35 48)(36 47)(37 46)(38 45)(39 44)(40 43)(41 42)(55 68)(56 67)(57 66)(58 65)(59 64)(60 63)(61 62)(69 81)(70 80)(71 79)(72 78)(73 77)(74 76)(82 88)(83 87)(84 86)(89 108)(90 107)(91 106)(92 105)(93 104)(94 103)(95 102)(96 101)(97 100)(98 99)(109 122)(110 121)(111 120)(112 119)(113 118)(114 117)(115 116)(123 135)(124 134)(125 133)(126 132)(127 131)(128 130)
G:=sub<Sym(135)| (1,129,85,75,28)(2,130,86,76,29)(3,131,87,77,30)(4,132,88,78,31)(5,133,89,79,32)(6,134,90,80,33)(7,135,91,81,34)(8,109,92,55,35)(9,110,93,56,36)(10,111,94,57,37)(11,112,95,58,38)(12,113,96,59,39)(13,114,97,60,40)(14,115,98,61,41)(15,116,99,62,42)(16,117,100,63,43)(17,118,101,64,44)(18,119,102,65,45)(19,120,103,66,46)(20,121,104,67,47)(21,122,105,68,48)(22,123,106,69,49)(23,124,107,70,50)(24,125,108,71,51)(25,126,82,72,52)(26,127,83,73,53)(27,128,84,74,54), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81)(82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135), (2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,54)(30,53)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)(61,62)(69,81)(70,80)(71,79)(72,78)(73,77)(74,76)(82,88)(83,87)(84,86)(89,108)(90,107)(91,106)(92,105)(93,104)(94,103)(95,102)(96,101)(97,100)(98,99)(109,122)(110,121)(111,120)(112,119)(113,118)(114,117)(115,116)(123,135)(124,134)(125,133)(126,132)(127,131)(128,130)>;
G:=Group( (1,129,85,75,28)(2,130,86,76,29)(3,131,87,77,30)(4,132,88,78,31)(5,133,89,79,32)(6,134,90,80,33)(7,135,91,81,34)(8,109,92,55,35)(9,110,93,56,36)(10,111,94,57,37)(11,112,95,58,38)(12,113,96,59,39)(13,114,97,60,40)(14,115,98,61,41)(15,116,99,62,42)(16,117,100,63,43)(17,118,101,64,44)(18,119,102,65,45)(19,120,103,66,46)(20,121,104,67,47)(21,122,105,68,48)(22,123,106,69,49)(23,124,107,70,50)(24,125,108,71,51)(25,126,82,72,52)(26,127,83,73,53)(27,128,84,74,54), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81)(82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135), (2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,54)(30,53)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)(61,62)(69,81)(70,80)(71,79)(72,78)(73,77)(74,76)(82,88)(83,87)(84,86)(89,108)(90,107)(91,106)(92,105)(93,104)(94,103)(95,102)(96,101)(97,100)(98,99)(109,122)(110,121)(111,120)(112,119)(113,118)(114,117)(115,116)(123,135)(124,134)(125,133)(126,132)(127,131)(128,130) );
G=PermutationGroup([[(1,129,85,75,28),(2,130,86,76,29),(3,131,87,77,30),(4,132,88,78,31),(5,133,89,79,32),(6,134,90,80,33),(7,135,91,81,34),(8,109,92,55,35),(9,110,93,56,36),(10,111,94,57,37),(11,112,95,58,38),(12,113,96,59,39),(13,114,97,60,40),(14,115,98,61,41),(15,116,99,62,42),(16,117,100,63,43),(17,118,101,64,44),(18,119,102,65,45),(19,120,103,66,46),(20,121,104,67,47),(21,122,105,68,48),(22,123,106,69,49),(23,124,107,70,50),(24,125,108,71,51),(25,126,82,72,52),(26,127,83,73,53),(27,128,84,74,54)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27),(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54),(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81),(82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108),(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135)], [(2,27),(3,26),(4,25),(5,24),(6,23),(7,22),(8,21),(9,20),(10,19),(11,18),(12,17),(13,16),(14,15),(29,54),(30,53),(31,52),(32,51),(33,50),(34,49),(35,48),(36,47),(37,46),(38,45),(39,44),(40,43),(41,42),(55,68),(56,67),(57,66),(58,65),(59,64),(60,63),(61,62),(69,81),(70,80),(71,79),(72,78),(73,77),(74,76),(82,88),(83,87),(84,86),(89,108),(90,107),(91,106),(92,105),(93,104),(94,103),(95,102),(96,101),(97,100),(98,99),(109,122),(110,121),(111,120),(112,119),(113,118),(114,117),(115,116),(123,135),(124,134),(125,133),(126,132),(127,131),(128,130)]])
75 conjugacy classes
| class | 1 | 2 | 3 | 5A | 5B | 5C | 5D | 9A | 9B | 9C | 10A | 10B | 10C | 10D | 15A | 15B | 15C | 15D | 27A | ··· | 27I | 45A | ··· | 45L | 135A | ··· | 135AJ |
| order | 1 | 2 | 3 | 5 | 5 | 5 | 5 | 9 | 9 | 9 | 10 | 10 | 10 | 10 | 15 | 15 | 15 | 15 | 27 | ··· | 27 | 45 | ··· | 45 | 135 | ··· | 135 |
| size | 1 | 27 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 27 | 27 | 27 | 27 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
75 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C5 | C10 | S3 | D9 | C5×S3 | D27 | C5×D9 | C5×D27 |
| kernel | C5×D27 | C135 | D27 | C27 | C45 | C15 | C9 | C5 | C3 | C1 |
| # reps | 1 | 1 | 4 | 4 | 1 | 3 | 4 | 9 | 12 | 36 |
Matrix representation of C5×D27 ►in GL2(𝔽271) generated by
| 187 | 0 |
| 0 | 187 |
| 231 | 203 |
| 68 | 163 |
| 183 | 259 |
| 171 | 88 |
G:=sub<GL(2,GF(271))| [187,0,0,187],[231,68,203,163],[183,171,259,88] >;
C5×D27 in GAP, Magma, Sage, TeX
C_5\times D_{27} % in TeX
G:=Group("C5xD27"); // GroupNames label
G:=SmallGroup(270,1);
// by ID
G=gap.SmallGroup(270,1);
# by ID
G:=PCGroup([5,-2,-5,-3,-3,-3,752,237,3003,138,4504]);
// Polycyclic
G:=Group<a,b,c|a^5=b^27=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export